1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

A curve has equation $$y = a \sin x + b \cos x$$ where \(a\) and \(b\) are constants. The maximum value of \(y\) is 4 and the curve passes through the point \(\left(\frac{\pi}{3}, 2\sqrt{3}\right)\) as shown in the diagram. \includegraphics{figure_6} Find the exact values of \(a\) and \(b\). [6 marks]

Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos (\theta + \alpha)$$ find the value of \(R\). Circle your answer. [1 mark] \(6\) \quad \(8\) \quad \(10\) \quad \(14\)

Express \(\cos\theta + \sqrt{3}\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R\) and \(\alpha\) are exact values to be determined. [4]

1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]

The temperature \(T °C\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5 \sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.

1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
2. Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 °C\). [6]

1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]

The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5\sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.

1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]

In this question you must show detailed reasoning.

1. Express \(8\cos x + 5\sin x\) in the form \(R\cos(x - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). [3]
2. Hence solve the equation \(8\cos x + 5\sin x = 6\) for \(0 \leqslant x < 2\pi\), giving your answers correct to 4 decimal places. [3]

\includegraphics{figure_5} A horizontal path connects an island to the mainland. On a particular morning, the height of the sea relative to the path, \(H\) m, is modelled by the equation $$H = 0.8 + k \cos(30t - 70)°$$ where \(k\) is a constant and \(t\) is number of hours after midnight. Figure 5 shows a sketch of the graph of \(H\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).

1. Find the time of day at which the height of the sea is at its maximum. [2] Given that the maximum height of the sea relative to the path is 2 m,
2. 1. find a complete equation for the model,
   2. state the minimum height of the sea relative to the path.
   [2] It is safe to use the path when the sea is 10 centimetres or more below the path.
3. Find the times between which it is safe to use the path. (Solutions relying entirely on calculator technology are not acceptable.) [4]

$$f(x) = 12 \cos x - 4 \sin x.$$ Given that \(f(x) = R \cos(x + \alpha)\), where \(R \geq 0\) and \(0 \leq \alpha \leq 90°\),

1. find the value of \(R\) and the value of \(\alpha\). [4]
2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leq x < 360°\), giving your answers to one decimal place. [5]
3. 1. Write down the minimum value of \(12 \cos x - 4 \sin x\). [1]
   2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. [2]